摘要
在管理科学、计算机科学、系统科学、信息科学以及工程应用等领域都存在着大量的不确定性,如随机性、模糊性、模糊随机性等.在模糊理论中,模糊集之间的距离是一个重要概念.在过去的几十年中提出了模糊集间距离的多种定义以反映模糊集的差别程度.模糊集之间的相似度被认为是距离的对偶,反映的是模糊集间的相近程度,也是模糊集理论的一个重要研究内容.不确定变量间的距离和相似度应用领域非常广泛,如：模式识别、机器学习、决策和市场预测等等.
本文分析了已有不确定环境下有关距离和相似度的研究成果,指出了存在的间题.在不确定理论框架下提出了不确定变量间的一类新距离和相似度,给出若干计算公式并进行证明.基于新距离对模糊过程的微积分进行了讨论,给出了模糊过程微积分的一些性质.本文工作进一步丰富了不确定理论的内容,具体研究内容如下：
提出了一类新的模糊变量之间、模糊随机变量之间、随机模糊变量之间和双重随机变量之间的距离,给出了相应的度量空间并证明了空间的完备性,讨论了这些距离的一些极限性质.进一步提出模糊向量之间、模糊随机向量之间、随机模糊向量之间和双重随机向量之间的距离及度量空间并讨论了有关性质.本文给出的有关距离完全满足距离度量的公理要求.
提出了模糊过程的概念,基于模糊变量间的新距离对模糊过程的连续性、模糊过程的微分和积分进行了讨论,给出了模糊过程微积分的一些性质.
基于可信性理论提出了模糊变量间的相似度,用于反映模糊变量之间的相近性,给出了多个计算相似度的公式.对模糊向量之间的相似度也进行了讨论,并举例说明相似度在模式识别中的应用.
在机会理论框架下给出了混合变量之间的相似度的概念及相应的计算公式和证明,讨论了混合向量之间的相似度并通过数值例子说明相似度的应用.
There exists a great deal of uncertainties such as randomness, fuzziness and fuzzyrandomness in the fields of management sciences, computer sciences, sys-tern sciences, information sciences, and engineering applications, etc. Distances between fuzzy sets is a important topic for fuzzy set theory. Various distances between fuzzy sets were presented to reflect the difference level of fuzzy sets in the last decades. Similarity measures between fuzzy sets can be regarded as a dual concept of distance measures reflecting] nearness of fuzzy sets. Distance measures and similarity measures between uncertain variables have been widely studied and applied in a variety of areas such as pattern recognition, machine learning, decision making and market prediction etc.
Concepts of distance and similarity of fuzzy sets given before are based on membership function on possibility measure, so they do not satisfy identification. In order to overcome this shortage, several definitions of distance measures and similarity measures between uncertain variables are put forward and correspond-ing proofs are given. Integral and differential of fuzzy processes based on new distances are discussed. The contents are described as follows:
A new kind of distances between fuzzy variables, fuzzy random variables, random fuzzy variables and birandom variables are proposed and these distances fully satisfy the mathematical axioms of a distance metric. Furthermore, metric spaces of different uncertain variables are defined, the completeness of this space is proved and the properties of new distances are discussed. Finally, the distances between fuzzy vectors, fuzzy random vectors, random fuzzy vectors and birandom vectors are also given.
The concept of continuity for fuzzy processes is proposed with our new dis-tances defined by the expected value operator and dealt with in the framework of credibility theory. Some properties of continuity for fuzzy processes are also proven. From this, the integral and differential of fuzzy processes are defined and their properties are discussed.
The definition of the degree of similarity between fuzzy variables is intro-duced to reflect nearness of fuzzy variables] Four similarity measures between fuzzy variables are proposed and corresponding proofs are given. Similarity mea-sures between fuzzy vectors and their calculational methods are also presented. The proposed similarity measures are applied to pattern recognition.
The definition of the degree of similarity between hybrid variables is in troduced based on chance theory. Several similarity measures between hybrid variables are proposed and corresponding proofs are given. Similarity measures between hybrid vectors are also discussed. Finally, a numerical example is given to illustrate the application of the proposed similarity measure.
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